## Contents |

## Introduction

John Smolin coined the phrase "Going to the Church of the Larger Hilbert Space" for the dilation constructions of channels and states, which not only provide a neat characterization of the set of permissible quantum operations but are also a most useful tool in quantum information science.

According to **Stinespring's dilation theorem**, every completely positive and trace-preserving map, or channel, can be built from the basic operations of (1) tensoring with a second system in a specified state, (2) unitary transformation, and (3) reduction to a subsystem. Thus, any quantum operation can be thought of as arising from a unitary evolution on a larger (*dilated*) system. The auxiliary system to which one has to couple the given one is usually called the ancilla of the channel. Stinespring's representation comes with a bound on the dimension of the ancilla system, and is unique up to unitary equivalence.

## Stinespring's dilation theorem

We present Stinespring's theorem in a version adapted to completely positive and trace-preserving maps between finite-dimensional quantum systems. For simplicity, we assume that the input and output systems coincide. The theorem applies more generally to completely positive (not necessarily trace-preserving) maps between *C*^{ * } − algebras.

**Stinespring's dilation:**Let be a completely positive and trace-preserving map between states on a finite-dimensional Hilbert space*H*. Then there exists a Hilbert space and a unitary operation*U*on such that- for all , where denotes the partial trace on the system.
- The ancilla space can be chosen such that . This representation is unique up to unitary equivalence.

## Kraus decomposition

It is sometimes useful not to go to a larger Hilbert space, but to work with operators between the input and output Hilbert spaces of the channel itself. Such a representation can be immediately obtained from Stinespring's theorem: We introduce a basis of the ancilla space and define the **Kraus operators** *t*_{k} in terms of Stinespring's unitary *U* as

The Stinespring representation then becomes the **operator-sum decomposition** or **Kraus decomposition** of the quantum channel *T*:

**Kraus decomposition:**Every completely positive and trace-preserving map can be given the form- for all . The Kraus operators satisfy the completeness relation .

## Purification of quantum states

Quantum states are channels with one-dimensional input space (cf. Channel (CP map)). We may thus apply Stinespring's dilation theorem to conclude that can be given the representation

- ,

where is a pure state on the combined system .
In other words, every mixed state can be thought of as arising from a pure state on a larger Hilbert space. This special version of Stinespring's theorem is usually called the **GNS construction** of quantum states, after Gelfand and Naimark, and Segal.

For a given mixed state with spectral decomposition , such a **purification** is given by the state

- .

## References and further reading

- M. A. Nielsen, I. L. Chuang:
*Quantum Computation and Quantum Information*; Cambridge University Press, Cambridge 2000 - K. Kraus:
*States, Effects, and Operations*; Springer, Berlin 1983 - E. B. Davies:
*Quantum Theory of Open Systems*; Academic Press, London 1976 - V. Paulsen:
*Completely Bounded Maps and Operator Algebras*; Cambridge University Press, Cambridge 2002 - M. Keyl:
*Fundamentals of Quantum Information Theory*; Phys. Rep.**369**(2002) 431-548; quant-ph/0202122 - W. F. Stinespring:
*Positive Functions on*; Proc. Amer. Math. Soc.*C*^{ * }− algebras**6**(1955) 211 - I. M. Gelfand, M. A. Naimark:
*On the Imbedding of Normed Rings into the Ring of Operators in Hilbert space*; Mat. Sb.**12**(1943) 197 - I. E. Segal:
*Irreducible Representations of Operator Algebras*; Bull. Math. Soc.**61**(1947) 69